CH.6 Compensation of control systems
Stability criterion in the Bode diagram
(1) When the magnitudefrequency curve crosses the 0dB line, if the corresponding phase angle is larger than 1800, then it’s stable. Vice versa. The crossing frequency with the 0dB line is called magnitude crossing frequency, denoted as ωc. (2) When the phasefrequency curve crosses the 1800 line, if the corresponding magnitude is negative, then it’s stable. Vice versa. The crossing frequency with the 0dB line is called phase crossing frequency, denoted as ωg.
20 dB
1
ω =7 16 c 5 10
1d 5B
50 100
ω
?
π
2
? π
3 π ? 2
γ =55
ω
ω g
The margin of magnitude : The margin of phase :
20 dB
1 Kg = G (jω ) H (jω )
ω =ωg
γ c = 180o + ? (ω c )
ω
1
ωc = 7 16 10 5
15dB
50
100
?
π
2
?π
? 3π 2
γ = 55
ω
ωg
To obtain a favorable performance, the magnitude margin should be larger than 6dB, while the phase margin should go between 3060°. °
Example 4：For a system: ：

K s ( s + 1)( s + 5 )
Calculate the magnitude margin and phase margin when K=10 and K=100. K=100
K=10 12dB 0dB +8dB
+21o 180
o
30o
61 Introduction
What’s compensation or correction of a control system ? K The OL of the system: G(s)H(s) = s(Ts +1)(s +1) How to make it stable. According to the Routh criterion, we can get:
T +1 1 K< =1+ (K > 0 T > 0) T T
But for the system:
K G(s)H(s) = 2 s (Ts +1)
According to Routh criterion, this system cannot be stable only varying K or T.
K(τ s +1) If we make: G(s)H(s) = 2 s (Ts +1)
This closedloop system can be stable.
τ >T
We make the system stable by adding a component. This procedure is called the compensation or correction. Definition of compensation: Adding a component, which makes the system’s performance to be improved, other than only by varying the system’s parameters, this procedure is called the compensation or correction of the system.
Compensator: The compensator is an additional component that is inserted into a control system to compensate for a deficient performance.
K Example: G(s)H(s) = 2 , to add an τs+1 component, s (Ts+1) the systemcan be stable, τs+1 is a compensator.
Types of the compensation
The transfer function of the compensator is designated as Gc (s), and according to the location of Gc (s) in the structure of the system, we can get several types:
(1) Cascade compensation (2) Feedback compensation (3) Both cascade and feedback compensation (4) Feedforward compensation
Cascade compensation
Features : simple but the effects to be restricted.
Feedback compensation
R(s)


G0 ( s ) GC
C(s)
R(s)
C(s)

G 10

G 20
GC
Features: complicated but noise limiting, the effects are better than the cascade compensation.
Both cascade and feedback compensation
R(s) C(s)

GC1

G0
GC 2
Features: have advantages both of cascade and feedback compensation.
Feedforward compensation
GC
R(s) F(s)
GC
+
C(s) R(s)

G10
G20

G10
+
C(s)
G20
For input For disturbance(voice) Features: theoretically we can make the error of a system to be zero and no effects to the transient performance of the system.
62 Operation analysis of the basic compensators
PID controller
Transfer function:
1 Gc(s) = Kp (1+ + τDs) τI s Kp 1 = Kp + KI + KDs → KI = ; KD = Kpτ D s τI
P proportional controller →promoting sensitivity I  integrating controller →clearing ess D differential controller →improving stability
PD controller
transfer function: Gc (s) = Kp + KDs
G (s) C
Assuming: G(s) =
R(s)＋
ω
s(s + 2ξωn )
2 n
－
Kp
KDs
＋ ＋
G(s)
C(s)
The open ? loop transfer function of the compensated
ωn2 (KP + KDs) system is: Gc (s)G(s) = s(s + 2ξωn )
It shows that the PD controller is equivalent to adding a KP open ? loop zero at : s = ? KD
Effects of PD controller: 1) PD controller does not alter the system type; 2) PD controller improve the system’s stability (to increase damping and reduce maximum overshoot); 3) PD controller reduce the rise time and settling time; 4) PD controller increase BW(Band Width) and improve GM(Kg), PM(γc) .
－ bring in the noise !
1 PI controller Transfer function : Gc (s) = K p + KI s R(s) ＋ ＋ C(s) Kp G(s) ωn2 － ＋ Assum : G(s) = ing 1 KI s(s + 2ξωn ) GC (s) s
The op ? loop transfer function of the com ensate d systemis : en p 1
ωn2(KP + KI ) ω 2(K s + K ) s = n P I Gc (s)G(s) = s(s + 2ξωn ) s2(s + 2ξωn )
It show that the PI controller is eq alent to ad ing a op ? loop s uiv d en KI zero at : s = ? and a p at : s = 0 ole KP
Effects of PI controller: 1) Increase the system’s type－clear the steadystate error ; － 2) reduce BW(Band Width) and GM(Kg), PM(γc); beneficial to the noise limiting , not beneficial to the system’s stability.
PID controller
R(s)
Transfer function: Gc(s) = K p + KI 1 + KDs s

Kp
KI 1 s
+
G(s)
C(s)
KDs
G C (s )
PID controller have advantages both of PI and PD.
Circuits of PID
R2 ur R1 C ur R1 R2
_
u0
_
u0
+ PI controller
R2 ur R1 C2
C
+ PD controller
_
u0
C1
+ PID controller
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